r3.07.04 Finite elements of solid shells

Summary:

In order to complete the library of finite elements for flat plates [R3.07.03] currently available in Code_Aster (DKT, DST, Q4G…), it is proposed to introduce two finite elements of a solid or three-dimensional shell [bib1]. This modeling COQUE_3D [U3.12.03] makes it possible to perform calculations of shell structures of any shape with a better approximation of geometry and kinematics.

We will limit ourselves to the framework of linear kinematics. We therefore remain in small movements and in small deformations. No restrictions are made on the type of behavior under plane constraints.

The two elements that are introduced are the quadratic quadratic quadrangle element Heterosis with 9 nodes and its triangular equivalent with 7 nodes. The formulation of the continuous problem is done in Cartesian coordinates, which makes it possible to avoid explicit calculations of curvatures. These two elements correspond to the 3-node shell line element presented in document [R3.07.02].

These two new elements are validated on existing plate test cases, and on three new shell test cases developed in the validation documentation and whose main conclusions are presented briefly here.

This note also shows in the appendix how to take into account the anisotropy of materials.

Table of Contents

• 1. Introduction
• 2. Formulation
  • 2.1. Shell geometry
  • 2.2. Plate and shell theory
• 3. Principle of virtual work
  • 3.1. Deformation work
  • 3.2. Work of forces and external couples
  • 3.3. Work of inertial forces
  • 3.4. Principle of virtual work
• 4. Numerical discretization of variational formulation based on the principle of virtual work
  • 4.1. Introduction
  • 4.2. Discretization of geometric terms
  • 4.3. Discretization of the displacement field
  • 4.4. Discretization of the deformation field
  • 4.5. Stiffness matrix
  • 4.6. Mass matrix
  • 4.7. Digital integration for elasticity
  • 4.8. Numerical integration for plasticity
  • 4.9. Discretization of basic work for loads
• 5. Validation
  • 5.1. Test case in linear statics
  • 5.2. Dynamic test case
• 6. Thermomechanical chaining
  • 6.1. Description
  • 6.2. Test case
• 7. Implanting shell elements in Code_Aster
  • 7.1. Description
  • 7.2. Use and developments introduced
  • 7.3. Linear elasticity calculation
  • 7.4. Plasticity calculation
• 8. Conclusion
• 9. Bibliography
• 10. Description of document versions